Tunable Bands in 1D Fractional Quantum Media

Abstract

Fractional calculus has become an essential framework in geophysics, optics, and biological systems to capture long-range correlations and anomalous transport. In this article, we extend this framework to explore a particle in a periodic potential, where the Schrodinger equation is extended to its fractional form. This allows us to study how the Levy index $q$ governs the formation and inversion of energy bands, offering a pathway to engineer new physical behaviors by tuning $q$ in periodic quantum systems. We solve the fractional Schrodinger equation (FSE) for periodic rectangular potentials of varying height $V_0$, barrier thickness $L$, and well width $W$ using an imaginary-time evolution algorithm, supplemented by Gaussian process regression. This analysis reveals a qualitative shift in the system's band structure at $q = 2$, separating into distinct regimes of dispersion for $q > 2$ and $q < 2$. For $q > 2$, the energy bands invert as symmetric minima emerge within the first Brillouin zone and shift from $k = 0$ toward $k=\pm \pi/a$ with increasing $q$. These degenerate minima define a Bloch-momentum qubit, suggesting an analog to valley degrees of freedom in valleytronics. The $q$ at which inversion completes scales as $q \propto V_0^{-0.28\pm0.05}$, $q \propto L^{-0.35\pm0.08}$, and $q \propto W^{-0.49\pm0.06}$ when varying parameters individually, indicating a tunable sensitivity to potential geometry. In contrast, for $q < 2$, the ground band hardens around $k = 0$, with a dispersion following $C|k|^q+E_0$ near $k = 0$. This suggests an effective mass of 0 for $1<q<2$ at the band's lowest energy state. These results demonstrate that the Levy index serves as a tunable degree of freedom in quantum periodic systems, capable of driving band inversion, modulating the band gap, and reshaping carrier dynamics through effective-mass control.